Chemotherapy and Total-Body Irradiation Followed by Donor Umbilical Cord Blood Transplant, Cyclosporine, and Mycophenolate Mofetil in Treating Patients With Hematologic Cancer

DISEASE CHARACTERISTICS: * Diagnosis of a hematologic malignancy of 1 of the following types: * Acute myeloid leukemia (AML), meeting the following criteria: * In complete remission (CR) by morphology (\< 5% blasts in the bone marrow), as defined by 1 of the following: * In first CR (CR1) and meets ≥ 1 of the following high-risk criteria: * High-risk cytogenetics (e.g., those associated with myelodysplastic syndromes \[MDS\] or complex karotype) * Preceding MDS * More than 2 courses of therapy was required to obtain CR * In second or greater CR * No morphologic relapse * Cytogenetic relapse or persistent disease allowed * Acute lymphocytic leukemia (ALL), meeting the following criteria: * In CR, as defined by 1 of the following: * In CR1 and meets ≥ 1 of the following high-risk criteria: * Unfavorable high-risk cytogenetics \[t(9;22), t(1;19), t(4;11) or other MLL rearrangements\] * More than 1 course of therapy was required to obtain CR * In second or greater CR * No morphologic relapse or persistent disease * Chronic myelogenous leukemia (CML), excluding refractory blast crisis * Advanced myelofibrosis * Advanced myelodysplasia (blasts \< 10% \[otherwise need AML induction pre-transplant\]), meeting ≥ 1 of the following criteria: * Refractory anemia with excess blasts (RAEB) * RAEB in transformation * Refractory anemia with severe pancytopenia * High-risk cytogenetics * Non-Hodgkin's lymphoma (NHL), meeting the following criteria: * One of the following histologic subtypes: * Mantle cell NHL * Disease progression after initial therapy (e.g., CHOP) * Beyond CR1 or beyond first partial remission (PR) * Intermediate-grade NHL in second or greater CR or PR * High-grade NHL * Stage III or IV disease AND received initial therapy * Stage I or II disease at diagnosis that subsequently progressed after a prior response duration of \< 1 year * No chemotherapy-refractory NHL (i.e., \< progressive disease after \> 2 salvage regimens) * Donor available, meeting the following criteria: * No other existing HLA-identical related donor available * 4-6/6 HLA-A, -B, and -DRB1, matched unrelated donor by molecular techniques * A and B to antigen level resolution * DR to allele resolution * Umbilical cord blood (UCB) graft may consist of one or two UCB units NOTE: A new classification scheme for adult non-Hodgkin's lymphoma has been adopted by PDQ. The terminology of "indolent" or "aggressive" lymphoma will replace the former terminology of "low", "intermediate", or "high" grade lymphoma. However, this protocol uses the former terminology. PATIENT CHARACTERISTICS: * Karnofsky score 80-100% (for adults) OR * Lansky score 50-100% (for children) * Creatinine ≤ 2.0 mg/dL (for adults) OR creatinine clearance \> 40 mL/min (for children) * Adults with a creatinine \> 1.2 mg/dL or a history of renal dysfunction must have a creatinine clearance \> 40 mL/min * Bilirubin ≤ 2 times normal * AST and ALT ≤ 2 times normal * Alkaline phosphatase ≤ 2 times normal * Pulmonary function \> 50 % of normal * LVEF ≥ 45% * No active infection, including Aspergillus or other mold, within the past 30 days * No history of HIV infection * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception PRIOR CONCURRENT THERAPY: * See Disease Characteristics * No prior myeloablative transplant within the past 6 months if ≤ 18 years old * No prior myeloablative allotransplant or autologous transplant if \> 18 years old * No prior extensive therapy (e.g., \> 12 months of alkylating therapy or \> 6 months of alkylating therapy with extensive radiation)